# Arrow-reversing Proj and blow-up

Let $$X$$ be a normal projective variety over the field of complex numbers. Let $$Y$$ be a subvariety of $$X$$, and let $$I_Y$$ be the ideal sheaf of $$Y$$. From what I know, I can define the blow-up of $$X$$ along $$Y$$ as the projective spectrum

$$Bl_YX=\operatorname{Proj} (\mathcal{O}_X\oplus I_Y \oplus I_Y^2 \oplus \ldots),$$

which comes with a surjective morphism $$\phi:Bl_YX\to X$$ (isomorphism everywhere except on $$Y$$).

The question is the following: the surjective morphism $$\phi$$ comes as an arrow-reversing map at the level of finitely generated, $$\mathbb{Z}$$-graded algebras? Since $$X$$ is projective, I can write $$X=\operatorname{Proj}(\bigoplus_{m\geq 0} H^0(X,\mathcal{O}_X(m)))$$, but

$$\bigoplus_{m\geq 0} H^0(X,\mathcal{O}_X(m)) \hookrightarrow \bigoplus_{m\geq 0} I_Y^m$$

doesn't look as an inclusion of algebras. I apologize for the probably dumb question but I'm having an hard time understanding the Proj.

The map you want isn't quite what you've written down: for instance, if $$X=\Bbb P^2$$ and $$Y$$ is a point, the LHS is $$k[x,y,z]$$ while the RHS is just $$k$$. Instead, you want to apply $$\Gamma_*(-)=\bigoplus_d \Gamma(-(d))$$ to the inclusion of $$\mathcal{O}_X$$ in to $$\bigoplus_m I_Y^m$$. The resulting map is injective because $$\mathcal{O}_X$$ is a direct summand of $$\bigoplus_m I_Y^m$$.
• Dear @KReiser, thank you for answering the question! Before accepting it, I have a few questions, as I don't understand the second part. Why would you want to apply $\Gamma_*$ (can you provide a reference?), and what you mean by $\Gamma(-(d))$ (some sort of shift)? Moreover, why do you want to consider the inclusion of $\mathcal{O}_X$? I warmly thank you again for the patience, have a nice day! Commented Aug 27, 2021 at 12:03
• The reason you want to apply $\Gamma_*$ to the inclusion $\mathcal{O}_X\to\bigoplus_m I_Y^m$ is because that gives you an injective map from $\bigoplus_{m\geq 0} H^0(X,\mathcal{O}_X(m))$ to $\bigoplus_{m\geq 0} H^0(X,\bigoplus_d I_Y^d)$, which is the correct version of the statement you want. By $\Gamma(-(d))$ I mean the functor that takes a sheaf $\mathcal{F}$ on $X$ to the global sections of $\mathcal{F}(d)$ - this is the same as taking $H^0(X,\mathcal{F}(d))$, which you're already doing. If you're looking for more material on $\Gamma_*$, check Hartshorne chapter II section 5, for instance. Commented Aug 27, 2021 at 18:31